Answer:
The options are missing, so I will give a general solution for this problem.
The first thing we need to do is describe the transformations used, so we can know how they impact the graph of f(x)
Vertical translation.
For a function f(x), a vertical translation of N units is written as:
g(x) = f(x) + N
If N is positive, then the translation is upwards
If N is negative, then the translation is downwards.
Horizontal translation
For a function f(x), a horizontal translation of N units is written as:
g(x) = f(x + N)
If N is positive, the translation is to the left.
If N is negative, the translation is to the right.
Vertical dilation.
For a function f(x), a vertical dilation of scale factor K, is written as:
g(x) = K*f(x)
Reflection across the x-axis.
For a function f(x), a reflection across the x-axis is written as:
g(x) = -f(x)
Now that we know how the transformations are used, let's analyze our problem:
We have f(x) replaced by g(x) = -3*f(x + 2) - 1
Let's construct g(x) and see all the transformations used.
So we start with g(x) = f(x) and keep adding transformations.
We can start with a vertical dilation of scale factor 3, then we get:
g(x) = 3*f(x)
(this will make our function steeper)
Now we can add a reflection across the x-axis:
g(x) = -3*f(x)
(this reflects the function across the x-axis)
Now we can translate the function 1 unit downwards:
g(x) = -3*f(x) - 1
(this moves the whole graph of the function one unit downwards)
Now we can translate the function 2 units to the left:
g(x) = -3*f(x + 2) - 1
(this moves the whole graph two units to the left)
Then the statement that describes this can be something like:
"A vertical dilation of scale factor 3, followed by a reflection across the x-axis, followed by a translation down of one unit, followed by a translation to the left of 2 units"
(The transformations could be applied in other order, this is not the only possible combination of transformations that transforms f(x) into g(x) )
